Meta

Category: Mathematics

For anyone who has an interest in sieve methods, I would like to point out that my new book

is now available from Cambridge University Press (and fairly extensive preview is found on Google books)… In fact, I’ve just received my first copy.

Of course, even before it appeared (but too late to incorporate the changes), I had found a few typos and mistakes, so a list of corrections is necessary and available; it will be updated as needed.

If you’re not quite sure if you are interested in sieve, I suggest a short look at this guest post on T. Tao’s blog (as well as Tao’s own discussion of the parity problem, though the latter is concerned with the so-called “small” sieves, which have a distinct flavor compared with the “large” sieve which I consider in the book).

Suppose there existed a natural probability (or density, or weakening of such) on the “set” of all models of the Zermelo-Fraenkel set-theory axioms. Suppose some “natural” mathematical statement had the property of having positive probability, different from 1, of holding in a random model. How should we interpret such a situation? Say, if the Continuum Hypothesis has probability 6/π2, of being false?

And if some natural statement P was shown to be a consequence of two other statements, having probability p and q, respectively, of holding in a random model, with p+q>1… so that the existence of a model where P holds would follow in highly non-constructive fashion… What would you think, philosophically or intuitively, of the “truth” of that statement?

When teaching the most elementary courses at the university, a little bit of combinatorics enters, and the relation between the binomial coefficients and the expansion of (x+y)k, for non-negative integers k. An often appreciated trick for the students is to prove some identities among binomial coefficients using “analytic” properties of polynomials, and interpret them combinatorially, or conversely. The most basic of these identities is probably

In the spirit of fun, assume we think of rewriting this as

And now, maybe while whistling idly to pretend that we are not doing anything, let’s jolt the denominators with a quick flick of the finger:

Since it seems that no one has noticed anything, let’s do it again:

and again, and again… but stop! a red-faced policeman comes, and says that he has nothing against some good clean fun, especially on Boat Race Night, but enough is enough, and what horror have we done with poor Newt’s lovely identity:

Well, of course, what is behind this is an undoubtedly well-known identity, which can be expressed in terms of hypergeometric functions with very general parameters. But the sliding denominators might be a nice thing to show students, and the (or at least, one elementary one) actual proof of the actual formula is a good exercise in exploiting “polynomiality” in various forms, so it can be used for that purpose…

So after re-reading carefully Zazie au pot de thèse, I have counted 74 references to mathematical terminology, including names of actual mathematicians (a few of which are actually hidden in puns which can be considered as Joycean or atrocious, depending on the point of view), and words which are used for their mathematical meaning in the text; this seems fair enough since, after all, the story is supposed to happen during (or mostly after) a mathematical PhD defence.

The best-hidden name (the construction being utterly untranslatable) is in the following sentence:

(Litterally: We must preserve the most minute of our obsolete habits, since we, irreducible Gauls, without customs, would lose our esprit de corps, our fundamental unity.)

The mathematician here is Galois, seen as “Gaulois-without-us”; the surroundings of field and Galois-theoretic wordings (irreducible, fundamental unit, field – which is the translated “corps” in French) were supposed to make this noticeable…

My favorite French novel is “Zazie dans le métro”, by Raymond Queneau. I won’t go into detailed literary criticism to explain why, though the amazing inventivity of the language is one reason, but mention only that the fact that Queneau is well-known to have had a lively interest in mathematics is certainly another factor (there are not many mathematical traces in this book, though there is a very nice sentence, which I can’t locate at the moment, which mixes elliptic, parabolic and hyperbolic…).

Parenthetically, Queneau’s interest was shared by all the writers of the OULIPO group, the best known of whom is probably G. Perec. Perec is the author of the book-without-e “La disparition”, but he also wrote an absolutely hilarious pastiche of a scientific paper (not a mathematical one) entitled “Cantratrix Sopranica L.” (which you really should read if you have never seen it; it’s in English). Even the bibliography is a jewel, as shown by the citation of a paper by “Einstein, Zweistein, Dreistein, Vierstein et Saint Pierre”. Note that this mathematical interest actually makes plausible the claim (which I have just seen on the web) that Grothendieck played a small role in L. Malle’s film version of “Zazie…”.

Coming back to “Zazie…”, when, to celebrate a friend’s PhD defense in Orsay in 1998, I decided to write a short story with the intention of cramming it with as many mathematical terms as I could in a non-professional context, I chose to imitate Queneau’s masterpiece, and make use of his characters Zazie, of the famously free vocabulary, and her uncle Gabriel, amateur of “sirop de grenadine” and, in the evening, dancer in a transvestite cabaret in Paris.

The resulting text is here; I will leave it as an exercise to find all the hidden mathematical terminology. The counts you may reach might not be the same as mine (which I will include in a follow-up after re-reading the text carefully, since I don’t remember it…). For instance, I don’t think I knew that “net” was a terme de métier when I wrote this text, and other instances yet unknown to me might lurk in it. As a teaser, I will mention “Eh quoi, si on élit P. Tique…”, which really must be read “Équation elliptique” (elliptic equation), and “Et les uns poussaient en avant, et les autres tiraient en arrière” (and some pushed forward, and some pulled back), which should be self-explanatory.